QTA 5 - SAMPLE MOMENTS

Question 1

What is the formula for estimating the mean?

Answer 1

𝜇̂ = (1/n) ∑ X𝑖 for i=1 to n

The mean estimator is based on independent and identically distributed random variables

Question 2

What is the difference between an estimator and an estimate?

Answer 2

An estimator is a function that calculates an estimate based on data, while an estimate is the value produced by applying the estimator to data

Question 3

What does the bias of an estimator measure?

Answer 3

The bias measures the difference between the expected value of the estimator and the true population value

Question 4

What does it mean that the mean estimator is BLUE?

Answer 4

BLUE stands for Best Linear Unbiased Estimator, indicating it has the lowest variance among linear unbiased estimators

Question 5

Define consistency of an estimator.

Answer 5

An estimator is consistent if it converges in probability to the true value as the sample size increases

Question 6

How do the Law of Large Numbers (LLN) and Central Limit Theorem (CLT) apply to the sample mean?

Answer 6

LLN states that the sample mean converges to the population mean as sample size increases, while CLT states that the distribution of the sample mean approaches a normal distribution as sample size increases

Question 7

What are skewness and kurtosis used to estimate?

Answer 7

Skewness measures the asymmetry of a distribution, while kurtosis measures the tail heaviness of the distribution

Question 8

What is the formula for estimating the variance?

Answer 8

𝜎^2 = (1/n) ∑ (X𝑖 - 𝜇̂)² for i=1 to n

The sample variance is a biased estimator

Question 9

What is the bias of the sample variance?

Answer 9

The bias is E[𝜎^2] - 𝜎² = -𝜎²/n

The sample variance tends to underestimate the population variance

Question 10

What is the relationship between standard deviation and standard error?

Answer 10

Standard deviation measures uncertainty in a data set, while standard error measures uncertainty of an estimator and decreases with increasing sample size

Question 11

How are annualized means and standard deviations calculated?

Answer 11

Annualized mean = monthly mean × 12, weekly mean × 52, daily mean × number of trading days per year

Standard deviations are scaled using the square root of the same factors

Question 12

What is the significance of log returns in finance?

Answer 12

Log returns allow for the summation of consecutive returns and are convenient for analysis

Question 13

What is the impact of sampling frequency on skewness?

Answer 13

Skewness does not follow a simple scaling law across sampling frequencies and may change direction at different frequencies

Question 14

What defines excess kurtosis in financial data?

Answer 14

Excess kurtosis indicates that the data has heavier tails than a normal distribution, typically seen in financial returns

Question 15

What are histograms used for?

Answer 15

Histograms represent the frequency distribution of a data series by dividing the data range into bins and counting occurrences

Question 16

How does a kernel density plot differ from a histogram?

Answer 16

A kernel density plot uses a weighted count of observations to provide a smooth estimate of the distribution, unlike the discrete bins of a histogram

Question 17

What is a linear estimator of the mean expressed as?

Answer 17

𝜇̂ = ∑ 𝑤𝑖X𝑖 for i=1 to n, where 𝑤𝑖 are weights that do not depend on X𝑖

Question 18

What does BLUE stand for in statistical estimation?

Answer 18

Best Linear Unbiased Estimator

Question 19

What property does a BLUE mean estimator have?

Answer 19

It has the lowest variance among all linear unbiased estimators.

Question 20

In the sample mean estimator, what are the weights (𝑤𝑖)?

Answer 20

𝑤𝑖 = 1/n

Question 21

What does the Law of Large Numbers (LLN) establish?

Answer 21

It establishes the large sample behavior of the mean estimator.

Question 22

What is the Kolmogorov Strong Law of Large Numbers?

Answer 22

It states that the average of 𝑖𝑖𝑑 random variables converges almost surely to the expected value.

Question 23

What is the significance of consistency in estimators?

Answer 23

An estimator is consistent if its finite sample bias diminishes as sample size increases.

Question 24

What happens to the variance of a consistent estimator as sample size increases?

Answer 24

The variance converges to zero.

Question 25

True or False: The sample mean is normally distributed for large sample sizes when data are 𝑖𝑖𝑑 normally distributed.

Answer 25

True

Question 26

What does the Central Limit Theorem (CLT) state about the sample mean estimator?

Answer 26

It states that the sample mean estimator is approximately normally distributed for large samples.

Question 27

What is the formula for the sample covariance estimator?

Answer 27

𝜎^𝑋F = (1/(𝑛 - 1)) ∑(X𝑖 − 𝜇̂𝑋)(𝑌𝑖 − 𝜇̂F)

Question 28

What is the relationship between covariance and correlation?

Answer 28

Correlation is the standardized version of covariance.

Question 29

How is the correlation coefficient estimated?

Answer 29

𝜌^𝑋F = 𝜎^𝑋F / (𝜎^𝑋 * 𝜎^F)

Question 30

What is the median in a distribution?

Answer 30

The 50% quantile where probabilities of values above or below it are equal.

Question 31

How is the median estimated from sorted data when the sample size is odd?

Answer 31

median(x) = x[n + 1/2]

Question 32

What is the formula for estimating the interquartile range (IQR)?

Answer 32

IQR = q^75 - q^25

Question 33

Why are quantiles considered robust to outliers?

Answer 33

They are unaffected by extreme values, unlike the mean.

Question 34

What does the CLT imply about the distribution of the sample mean?

Answer 34

It implies that the distribution of the sample mean is centered on the population mean and variance declines as sample size increases.

Question 35

What is the definition of covariance?

Answer 35

Covariance measures the linear dependence between two random variables.

Question 36

What does the term 'almost surely' mean in the context of convergence?

Answer 36

It indicates that the convergence occurs with probability 1.

Question 37

What happens to the distribution of the sample mean as sample size increases?

Answer 37

The distribution becomes narrower and more concentrated around the population mean.

Question 38

What is the relationship between the sample means of two variables and the CLT?

Answer 38

The CLT can be applied to the joint behavior of the two mean estimators as a bivariate statistic.

Question 39

What is the significance of the 2-by-2 covariance matrix in the context of bivariate data?

Answer 39

It is used to describe the joint distribution of the mean estimators for two variables.

Question 40

What does the bivariate Central Limit Theorem (CLT) imply about correlation in data?

Answer 40

Correlation in the data produces a correlation between the sample means, which is identical to the correlation between the data series.

Question 41

Define coskewness in the context of random variables.

Answer 41

Coskewness measures the likelihood of one variable taking a large directional value whenever the other variable is large in magnitude.

Question 42

How many different measures are there when computing cross p-th moments?

Answer 42

There are p - 1 different measures.

Question 43

List the types of cross moments when applying the principle to the first four moments.

Answer 43

* No cross means * One cross variance (covariance) * Two measures of cross-skewness (coskewness) * Three cross-kurtoses (cokurtosis)

Question 44

What is the significance of coskewness being zero in a bivariate normal distribution?

Answer 44

The coskewness is always 0, indicating no sensitivity to the direction of one variable to the magnitude of the other.

Question 45

Fill in the blank: The two coskewness measures are represented as _______.

Answer 45

[E s(X, X, Y), E X^2 Y - E X E Y]

Question 46

What does cokurtosis measure in relation to two series?

Answer 46

Cokurtosis captures the sensitivity of the magnitude of one series to the magnitude of the other series.

Question 47

What are the configurations of the three measures of cokurtosis?

Answer 47

* (1,3) * (2,2) * (3,1)

Question 48

True or False: The (2,2) cokurtosis measure is the easiest to understand.

Answer 48

True

Question 49

How is the (2,2) cokurtosis measure defined?

Answer 49

It is defined as capturing the sensitivity of the magnitude of one series to the magnitude of the other series.

Question 50

What does a large (2,2) cokurtosis indicate?

Answer 50

It indicates that both series tend to be large in magnitude at the same time.

Question 51

List the variables for which the sample analog of coskewness can be estimated.

Answer 51

* X * Y

Question 52

What is the formula for estimating coskewness?

Answer 52

ŝ X, X, Y = (1/n) ∑ (X_i - μ̂_X)(Y - μ̂_Y)

Question 53

What does the cokurtosis use in its definition?

Answer 53

It uses combinations of powers that add to 4.

Question 54

List the variables involved in the cokurtosis measures.

Answer 54

* k(X, X, Y, Y) * k(X, Y, Y, Y) * k(X, X, X, Y)

Question 55

What is the relationship between the sample means and their limiting distribution in practice?

Answer 55

Mean estimators are treated as if they are normally distributed.

Question 56

What are the two measures of coskewness?

Answer 56

* E s(X, X, Y) * E X^2 Y - E X E Y

Question 57

What is the significance of the variance in the context of coskewness?

Answer 57

Coskewness is standardized by the variance of one of the variables and the standard deviation of the other.

Question 58

Fill in the blank: The univariate skewness estimators are s(X, X, X) and _______.

Answer 58

s(Y, Y, Y)